CYK algorithm

The Cocke-Younger-Kasami (CYK) algorithm (alternatively called CKY) determines whether a
string can be generated by a given context-free grammar and, if so, how it can be generated. This is known as parsing the string. The algorithm is an example of dynamic programming.
The standard version of CYK recognizes languages defined by context-free grammars written in Chomsky normal form (CNF). Since any context-free grammar can be converted to CNF without too much difficulty, CYK can be used to recognize any context-free language. It is also possible to extend the CYK algorithm to handle some context-free grammars which are not written in CNF; this may be done to improve performance, although at the cost of making the algorithm harder to understand.
The worst case asymptotic time complexity of CYK is Θ(n³), where "n" is the length of the parsed string. This makes it one of the most efficient (in those terms) algorithms for recognizing any context-free language. However, there are other algorithms that will perform better for certain subsets of the context-free languages.
The algorithm
The CYK algorithm is a bottom up algorithm and is important theoretically, since it can be used to constructively prove that the membership problem for context-free languages is decidable.
The CYK algorithm for the membership problem is as follows: Let the input string consist of "n" letters, "a"₁ ... "a"_"n". Let the grammar contain "r" terminal and nonterminal symbols "R"₁ ... "R"_"r". This grammar contains the subset R_s which is the set of start symbols. Let P [n,n,r] be an array of booleans. Initialize all elements of P to false. For each i = 1 to n For each unit production R_j -> a_i, set P [i,1,j] = true. For each i = 2 to n -- Length of span For each j = 1 to n-i+1 -- Start of span For each k = 1 to i-1 -- Partition of span For each production R_A -> R_B R_C If P [j,k,B] and P [j+k,i-k,C] then set P [j,i,A] = true If any of P [1,n,x] is true (x is iterated over the set s, where s are all the indices for R_s) Then string is member of language Else string is not member of language
Informally
In informal terms, this algorithm considers every possible subsequence of the sequence of words and sets P [i,j,k] to be true if the subsequence of words starting from i of length j can be generated from R_k. Once it has considered subsequences of length 1, it goes on to subsequences of length 2, and so on. For subsequences of length 2 and greater, it considers every possible partition of the subsequence into two parts, and checks to see if there is some production P → Q R such that Q matches the first part and R matches the second part. If so, it records P as matching the whole subsequence. Once this process is completed, the sentence is recognized by the grammar if the subsequence containing the entire sentence is matched by the start symbol.
Algorithm extension
It is simple to extend the above algorithm to not only determine if a sentence is in a language, but to also construct a parse tree, by storing parse tree nodes as elements of the array, instead of booleans. Since the grammars being recognized can be ambiguous, it is necessary to store a list of nodes (unless one wishes to only pick one possible parse tree); the end result is then a forest of possible parse trees.An alternative formulation employs a second table B [n,n,r] of so-called "backpointers".
It is also possible to extend the CYK algorithm to parse strings using weighted and stochastic context-free grammars. Weights (probabilities) are then stored in the table P instead of booleans, so P [i,j,A] will contain the minimum weight (maximum probability) that the substring from i to j can be derived from A. Further extensions of the algorithm allow all parses of a string to be enumerated from lowest to highest weight (highest to lowest probability).